Problem 9 - Precalculus

Question

Use degree measure for your answers. In parts (c) and (d), use a calculator and round the results to one decimal place. (a) In \(\triangle A B C, \sin B=\sqrt{2} / 2 .\) What are the possible values for \(\angle B ?\) (b) In \(\triangle D E F, \cos E=\sqrt{2} / 2 .\) What are the possible values for \(\angle E ?\) (c) In \(\triangle G H I, \sin H=1 / 4 .\) What are the possible values for \(\angle H ?\) (d) In \(\triangle J K L, \cos K=-2 / 3 .\) What are the possible values for \(\angle K ?\)

Step-by-Step Solution

Answer

(a) 45°, 135°; (b) 45°, 315°; (c) 14.5°, 165.5°; (d) 131.8°, 228.2°.

Step 1: Analyze Part (a) - Given Trigonometric Value

We need to find the angles where \(\sin B = \frac{\sqrt{2}}{2}\). The basic angles for which this is true are \(45^\circ\) and \(135^\circ\).

Step 2: Determine All Possible Angles for \(\angle B\)

For sine, the angle can either be in the first or second quadrants where sine is positive. Thus, the possible values for \(\angle B\) are \(45^\circ\) and \(135^\circ\).

Step 3: Analyze Part (b) - Given Trigonometric Value

We need to find the angles where \(\cos E = \frac{\sqrt{2}}{2}\). The general angles for cosine with this value are \(45^\circ\) and \(315^\circ\).

Step 4: Determine All Possible Angles for \(\angle E\)

For cosine, it is positive in the first and fourth quadrants. Hence, the possible values for \(\angle E\) are \(45^\circ\) and \(315^\circ\).

Step 5: Solve Part (c) - Use a Calculator for \(\angle H\)

Now, find the angle \(\angle H\) where \(\sin H = \frac{1}{4}\). Use a calculator to get \(\angle H \approx 14.5^\circ\) in the first quadrant.

Step 6: Consider Sine's Symmetry for \(\angle H\)

Since sine is also positive in the second quadrant, find the second angle by computing \(180^\circ - 14.5^\circ = 165.5^\circ\). Thus, possible angles for \(\angle H\) are \(14.5^\circ\) and \(165.5^\circ\).

Step 7: Solve Part (d) - Use a Calculator for \(\angle K\)

For \(\cos K = -\frac{2}{3}\), use a calculator to find \(\angle K\). The angle in the second quadrant is \(\angle K \approx 131.8^\circ\).

Step 8: Consider Cosine's Negative Value for \(\angle K\)

Since cosine is negative in the second and third quadrants, the third-quadrant angle is \(360^\circ - 131.8^\circ = 228.2^\circ\). Thus, possible angles for \(\angle K\) are \(131.8^\circ\) and \(228.2^\circ\).

Key Concepts

Degree Measure

Degrees are a way to measure angles. This system divides a full circle into 360 equal parts. Each part is a degree. Understanding degrees helps us navigate geometry, trigonometry, and real-world angles. To understand this measure better, think of a clock. A full rotation of the clock's hands is 360 degrees. This helps when determining angles in triangles and other shapes. Each triangle's angles combined always equal 180 degrees. This fact is vital for solving trigonometry problems involving angles. In exercises like this, answers are often expressed in degrees. Calculators can easily switch to degree mode for precise results.

Sine Function

The sine function is key in trigonometry. It relates an angle in a right triangle to the lengths of the triangle's sides. Specifically, it is the ratio of the opposite side to the hypotenuse. The sine function varies, reaching its maximum of 1 and minimum of -1. In solving angle problems:

• Sine is positive in both the first and second quadrants.
• Given \( \sin B= \frac{\sqrt{2}}{2} \), possible angles are \( 45^\circ \) and \( 135^\circ \).
• Given \( \sin H=\frac{1}{4} \,\) calculate using inverse sine function with a calculator for precision.

Identifying angles with sine involves considering symmetry properties in different quadrants.

Cosine Function

The cosine function also plays a fundamental role in trigonometry. It describes the variation of the cosine of an angle in terms of the lengths of triangle sides. Specifically, it is the ratio of the adjacent side to the hypotenuse. The cosine function value is:

• Positive in the first and fourth quadrants.
• Negative in the second and third quadrants.

For example, given \( \cos E=\frac{\sqrt{2}}{2} \,\) the angles are \( 45^\circ \) and \( 315^\circ \.\) With \( \cos K=-\frac{2}{3} \,\) calculating with a calculator gives approximately \( 131.8^\circ \) in the second quadrant, applying symmetry gives the possible angles in the third quadrant too. Explore both positive and negative values for an understanding of angle determination.

Angle Determination

Determining angles using trigonometric functions often requires calculating potential angles. These calculations involve:

• Recognizing quadrant-specific behaviors of sine and cosine functions.
• Using inverse trigonometric functions on a calculator to compute specific angle values to obtain precise results.

For angles like those in the exercises, note both positive and negative scenarios:

• For \( \sin H = \frac{1}{4} \,\) calculate quadrant one and the related angle \( 180^\circ - 14.5^\circ \).
• For \( \cos K = -\frac{2}{3} \,\) consider quadrant two and three, derived from the calculator result. Symmetrically analyzing negative values provides additional angles.

Recognizing symmetry in trigonometric function values also helps students correctly identify all possible angles, ensuring a thorough understanding.